The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions

We consider the spectral Dirichlet problem for the Laplace operator in the plane Ω∘ with double-periodic perforation but also in the domain Ω• with a semi-infinite foreign inclusion so that the Floquet–Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra σ∘ and σ• of the problems in Ω∘ and Ω•, namely we present a concrete geometry which supports the relation σ∘⫋σ• due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium.